The category of compact metric spaces and its functional analytic duals

Abstract

A Lipschitz algebra Lip(X,dx) over a compact metric space (X,dx) consists of all complex valued continuous functions on (X,dx) which are Lipschitz with respect to dx and the standard metric on the complex plane C (absolute value). The norm on Lip(X,dx) is given by ||/|| = sup{|/(x)| : x e X} + sup{|/(z) - f(y)\/dx(x,y) : x,y G X & x + y}. We show that the category CLip in which objects are Lipschitz algebras and morphisms are algebra homomorphisms is dual to the category CMet in which objects are compact metric spaces and morphisms are Lipschitz maps. Let (X, d) be any metric space, and let Y = {(x,y) e X x X : x ^ y}. De Leeuw derivation defined by the metric d is the operator D : Cf,(X) -> Ct,(Y) be defined by (Df)(x,y) = (f(y) - f(x))/d(x,y) for (x,y) G Y. We consider the category CDer in which objects are pairs (C(X), DX), where (X, dx) is a compact metric space and Dx is the corresponding de Leeuw derivation, and morphisms are all homomorphisms v : C(X) -> C(Y) for which / 6 Domain(Dx) implies vf e Domain(Dy). We show that CDer is equivalent to CLip, and that CDer is dual to CMet.